at least one display device;

at least one input device;

at least one processor; and

at least one memory device which stores a plurality of instructions, which when executed by the at least one processor, cause the at least one processor to operate with the at least one display device and the at least one input device, after a random occurrence of a triggering event, for each play of a game, to:

(a) cause a display of at least two independently operable indicators, each of the indicators including a plurality of different, individual components,

(b) cause a display of a table, the table including:

(i) a separate display of each and every possible combination of the different components, and

(ii) for each of the displayed combinations of the different components, a separate display of an award offer associated with the displayed combination of the different components, each one of the displayed award offers being greater than zero prior to any generations of any of the displayed indicators, and a plurality of the displayed award offers being different,

(c) cause a display of a designated number of generations of the displayed indicators, said designated number being at least two,

(d) for each of the displayed indicators, independently randomly generate one of the different components for said displayed indicator, the randomly generated components generating one of the displayed combinations, wherein each one of the displayed combinations can be generated for each one of the generations,

(e) after randomly generating one of the different components for each of said displayed indicators, cause a display of the award offer associated with the generated combination of the randomly generated components such that the displayed award offer is determined independently of any other displayed award offers in any other of the generations,

(f) if each of the displayed indicators have generated the designated number of generations, cause the displayed award offer to correspond to a final award offer,

(g) if each of the displayed indicators have generated less than the designated number of generations, receive an input from a player whether to accept or reject the displayed award offer,

(h) if the received input corresponds to the player rejecting the diplayed award offer:

(i) cause a display of a number of generations of the displayed indicators remaining, and

(ii) repeat (d) to (h), and

(i) if the received input corresponds to the player accepting the displayed award offer or if the displayed award offer corresponds to the final award offer, cause an award associated with the displayed award offer to be provided to the player.